Abstract
In this paper, we mainly investigate global solutions and blow-up phenomena for the Cauchy problem of a generalized Degasperis–Procesi equation. We develop a new approach to consider the equation for (1−∂x)u instead of (1−∂x2)u. We first prove a new conserved quantity ‖(1−∂x)u‖L1 and give the local characterization of this conserved quantity. These properties then allow us to improve considerably a previous global existence result in [38], and establish two new blow-up results for strong solutions to the equation. In the end, based on vanishing viscosity method and using the conserved quantity we deduce the existence of global weak solutions to the equation for any initial data belonging to L1∩BV.
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